Tensors, Relativity, and Cosmology

276 Pages·2015·2.48 MB·English

by  Mirjana Dalarsson

Tensors, Relativity,
and Cosmology
Tensors, Relativity,
and Cosmology
Second Edition
M. Dalarsson
EricssonResearchandDevelopment
Stockholm,Sweden
N. Dalarsson
RoyalInstituteofTechnology
Stockholm,Sweden
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TabolfCe o ntents
Preface
XI
1 Introduction 1
PAR1T TensAolrg ebra 3
2 Notatainodn SyosftN eummsb ers 5
2.1I ntrodauncBdta isCoioncn cepts 5
2.2S ymmeatnrAdin ct isymSmyesttreimcs 6
2.3O perawtiihtSo ynsst ems 7
2.4S ummatCioonnv ention 9
2.5U niSty mmeatnrAdin ct mimseytrSiycs tems 10
Vect5po arc es
3 13
3.1I ntrodauncBdta isCoioncn cepts 13
3.2D efiniotafiV oenc Stpoarc e 14
3.3T hEeu cdleiaMne tSrpiacc e 15
3.4T hRei emanSnpiaacne s 15
DefiniotfTi eonnsso rs 21
4
4.1T ransforomVfa atriioanbsl es 21
4.2C ontravVaercitaonrts 21
4.3C ovarVieacntto rs 22
4.4I nvar(iSacnatlsa rs) 22
4.5C ontravTaernisaonrts 23
4.6C ovarTieannsto rs 23
4.7M ixTeedn sors 24
V
vi Table of Contents
4.8 Symmetry Properties of Tensors 24
4.9 Symmetric and Antisymmetric Parts of Tensors 25
4.10 Tensor Character of Systems 26
5 Relative Tensors 29
5.1 Introduction and Definitions 29
5.2 Unit Antisymmetric Tensors 29
5.3 Vector Product in Three Dimensions 31
5.4 Mixed Product in Three Dimensions 33
5.5 Orthogonal Coordinate Transformations 34
6 The Metric Tensor 39
6.1 Introduction and Definitions 39
6.2 Associated Vectors and Tensors 41
6.3 Arc Length of Curves: Unit Vectors 43
6.4 Angles between Vectors 44
6.5 Schwarz Inequality 45
6.6 Orthogonal and Physical Vector Coordinates 46
7 Tensors as Linear Operators 49
PART 2 Tensor An alysis 53
8 Tensor Derivatives 55
8.1 Differentials of Tensors 55
8.2 Covariant Derivatives 58
8.3 Properties of Covariant Derivatives 60
8.4 Absolute Derivatives of Tensors 61
9 Christoffe l Symbols 63
9.1 Properties of Christoffel Symbols 63
9.2 Relation to the Metric Tensor 65
Table of Contents vii
10 Differential Operators 69
10.1 The Hamiltonian '\/-Operator 69
10.2 Gradient of Scalars 69
10.3 Divergence of Vectors and Tensors 70
10.4 Curl of Vectors 71
10.5 Laplacian of Scalars and Tensors 72
10.6 Integral Theorems for Tensor Fields 73
1 1 Geodesic Lines 77
11.1 Lagrange Equations 77
11.2 Geodesic Equations 79
12 The Curvature Tensor 83
12.1 Definition of the Curvature Tensor 83
12.2 Properties of the Curvature Tensor 85
12.3 Commutator of Covariant Derivatives 87
12.4 Ricci Tensor and Scalar 89
12.5 Curvature Tensor Components 89
PART 3 Special Theory of Relativity 93
13 Relativistic Kinematics 95
13.1 The Principle of Relativity 95
13.2 Invariance of the Speed of Light 95
13.3 The Interval between Events 96
13.4 Lorentz Transformations 99
13.5 Velocity and Acceleration Vectors 102
14 Relativistic Phenomena 105
14.1 Introduction 105
14.2 Time Dilatation 105
14.3 Length Contraction 106
viii Table of Contents
14.4 Relativistic Addition of Velocities 108
14.5 The Twin Paradox 109
15 Relativistic Dynamics 111
15.1 Lagrange Equati ons 111
15.2 Energy-Momen tum Vector 113
15.3 Angular Momentum Tensor 118
16 Electromagnetic Fields 121
16.1 Electrom agnetic Field Tensor 121
16.2 Gauge Invariance 125
16.3 Lorentz Transformat ions and Invariants 126
17 Electromagnetic Field Equations 131
17.1 Electromagne tic Current Vector 131
17.2 Maxwell Equations 133
17.3 Electrom agnetic Potentials 137
17.4 Energy-Momentum Tensor 138
PART 4 General Theory of Relativity 145
18 Gravitational Fields 147
18.1 Introduction 147
18.2 Time Intervals and Distances 148
18.3 Particle Dynamics 150
18.4 Electromagnetic Field Equati ons 153
19 Gravitational Field Equations 157
19.1 The Action Integral 157
19.2 Action for Matter Fields 162
19.3 Einstein Field Equations 166
Table of Contents ix
20 Solutions of Field Equati ons 171
20.1 The Newton Law 171
20.2 The Schwarzschild So]ution 172
21 Appli cations of the Schwarzschild Metri c 183
21.1 The Perihelion Advance 183
21.2 The Black Holes 190
22 Gravitational Waves 197
22.1 Introduc tion 197
22.2 Linearized Einstein Field Equations 197
22.3 Plane-wave Solutions 202
22.4 Effects of Gravitationa l Waves on Particles 205
22.5 Generation of Gravita tional Radiation 208
PART 5 Elements of Cosmology 211
23 The Robertson-Wal ker Metric 213
23.1 Introduction and Basic Observations 213
23.2 Metric Definit ion and Propert ies 214
23.3 The Hubble Law 220
23.4 The Cosmo logical Red Shifts 221
24 The Cosmic Dynami cs 225
24.1 The Einstein Tensor 225
24.2 Friedmann Equations 233
25 Non-stati c Models of the Universe 237
25.1 Solutions of Friedmann Equat ions 237
25.2 Closed or Open Universe 241
25.3 Newtonian Cosmology 242